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A153278 Array read by antidiagonals of higher order Fubini numbers.

%I A153278 #25 May 12 2023 15:30:26
%S A153278 1,1,3,1,4,13,1,5,23,75,1,6,36,175,541,1,7,52,342,1662,4683,1,8,71,
%T A153278 594,4048,18937,47293,1,9,93,949,8444,57437,251729,545835,1,10,118,
%U A153278 1425,15775,143783,950512,3824282,7087261,1,11,146,2040,27146,313920,2854261,17975438,65361237,102247563
%N A153278 Array read by antidiagonals of higher order Fubini numbers.
%H A153278 Alois P. Heinz, <a href="/A153278/b153278.txt">Antidiagonals n = 1..150, flattened</a>
%H A153278 Istvan Mezo, <a href="https://arxiv.org/abs/0812.4047">On powers of Stirling matrices</a>, arXiv:0812.4047 [math.CO], 2008.
%H A153278 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela, and A. I. Solomon, <a href="https://arxiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a>, arXiv:quant-ph/0312202, 2003; J.Phys. A: Math.Gen. 37 3475-3487 (2004).
%e A153278 The table on p.6 of Mezo begins:
%e A153278 ===========================================================
%e A153278 F_p,n|n=1|n=2|n=3.|.n=4.|..n=5.|....n=6.|.....n=7.|comment
%e A153278 ===========================================================
%e A153278 p=1..|.1.|.3.|.13.|..75.|..541.|...4683.|...47293.|.A000670
%e A153278 p=2..|.1.|.4.|.23.|.175.|.1662.|..18937.|..251729.|.A083355
%e A153278 p=3..|.1.|.5.|.36.|.342.|.4048.|..57437.|..950512.|.A099391
%e A153278 p=4..|.1.|.6.|.52.|.594.|.8444.|.143783.|.2854261.|.A363008
%e A153278 p=5..|.1.|.7.|.71.|.949.|15775.|.313920.|.7279795.|.A363009
%e A153278 ===========================================================
%p A153278 f:= proc(n) option remember; local k; if n<=1 then 1 else
%p A153278        add(binomial(n, k) *f(n-k), k=1..n) fi
%p A153278     end:
%p A153278 stirtr:= proc(a) proc(n) option remember;
%p A153278            add( a(k) *Stirling2(n,k), k=0..n)
%p A153278          end end:
%p A153278 F:= (p,n)-> (stirtr@@(p-1))(f)(n):
%p A153278 seq(seq(F(d-n, n), n=1..d-1), d=1..13); # _Alois P. Heinz_, Feb 02 2009
%t A153278 f[n_] := f[n] = If[n <= 1, 1, Sum[Binomial[n, k]*f[n-k], {k, 1, n}]];
%t A153278 stirtr[a_] := Module[{g}, g[n_] := g[n] = Sum[a[k]*StirlingS2[n, k], {k, 0, n}]; g];
%t A153278 F[p_, n_] := (Composition @@ Table[stirtr, {p-1}])[f][n];
%t A153278 Table[Table[F[d-n, n], {n, 1, d-1}], {d, 1, 13}] // Flatten (* _Jean-François Alcover_, Mar 30 2016, after _Alois P. Heinz_ *)
%Y A153278 Cf. A000670, A083355, A099391, A153277, A363008, A363009.
%K A153278 easy,nonn,tabl
%O A153278 1,3
%A A153278 _Jonathan Vos Post_, Dec 22 2008
%E A153278 More terms from _Alois P. Heinz_, Feb 02 2009